In a radio frequency (RF) receiving or transmitting system with an array of antennas for frequency-division multiplexed communications, when only one RF chain is available, entry-wise elements of the channel may not be accessible. Instead, in each time or frequency resource, a linear combination of the elements may be obtained. This may be the case, for example, in a system for millimeter-wave (mm-wave) communications, in which the high power consumption of mixed signal components, and the high cost of RF chains, may make it costly to realize digital baseband beamforming, of the kind that may be used in lower-frequency multiple-input multiple-output (MIMO) systems. In such a system for mm-wave communications, analog beamforming may instead be used; all the antennas (of the array) may share a single RF chain and have weights of the same amplitude, i.e., a constant-amplitude constraint may apply to their weights.
The linear combination may be obtained using a phase shifter vector that may be referred to as a beamforming codeword; a set of such vectors, one for each beam to be formed, may be referred to as a beamforming codebook. The codebook may be represented as an array, each column of the array being a codeword corresponding to a respective beamforming codeword.